Welcome to CS262: Computational Genomics

Transcription

1 Instructor: Welcome to CS262: Computational Genomics Serafim Batzoglou TA: Paul Chen Tuesdays & Thursdays 12:50-2:05pm Clark S361

2 Goals of this course Introduction to Computational Biology & Genomics Basic concepts and scientific questions Why does it matter? Basic biology for computer scientists In-depth coverage of algorithmic techniques Current active areas of research Useful algorithms Dynamic programming String algorithms HMMs and other graphical models for sequence analysis

3 Topics in CS262 Part 1: Basic Algorithms Dynamic Programming & sequence alignment HMMs, CRFs & sequence modeling Sequence indexing; Burrows-Wheeler transform, De Brujin graphs Part 2: Topics in computational genomics and areas of active research DNA sequencing and assembly Comparative genomics Human genome resequencing Alignment Compression Human genome variation Cancer genomics Functional genomics Population genomics

4 Course responsibilities Homeworks 4 challenging problem sets, 4-5 problems/pset Due at beginning of class Up to 3 late days (24-hr periods) for the quarter Collaboration allowed please give credit Teams of 2 or 3 students Individual writeups If individual (no team) then drop score of worst problem per problem set (Optional) Scribing Due one week after the lecture, except special permission Scribing grade replaces 2 lowest problems from all problem sets First-come first-serve, staff list to sign up

5 Reading material Main Reading: Lecture notes Papers Optional: Biological sequence analysis by Durbin, Eddy, Krogh, Mitchison Chapters 1-4, 6, 7-8, 9-10

6 Birth of Molecular Biology Phosphate Group Sugar Nitrogenous Base A, C, G, T DNA A G C G A C T T C G C T G A G C Physicist Ornithologist

7 Genetics in the 20 th Century

8 Human Genome Project 1990: Start 3 billion basepairs $3 billion 2000: Bill Clinton: 2001: Draft 2003: Finished now what? most important scien.fic discovery in the 20th century

9 Sequencing Growth Cost of one human genome 2004: $30,000, : $100, : $10, : $1,000???: $300 How much would you pay for a smartphone?

10 Uses of Genomes Medicine Prenatal/Mendelian diseases Drug dosage (eg. Warfarin) Disease risk Diagnosis of infections Ancestry Genealogy Nutrition? Psychology? Baby Engineering???... Ethical Issues

11 How soon will we all be sequenced? Cost Killer apps Roadblocks? Applications Cost 2015? 2020? Time

12 Intro to Biology

13 Sequence Alignment

14 Evolution CT Amemiya et al. Nature 496, (2013) doi: /nature12027

15 Evolution at the DNA level Deletion Mutation ACGGTGCAGTTACCA SEQUENCE EDITS AC----CAGTCCACCA REARRANGEMENTS Inversion Translocation Duplication

16 Evolutionary Rates next generation OK OK OK X X Still OK?

17 Sequence conservation implies function Alignment is the key to Finding important regions Determining function Uncovering evolutionary events

18 Sequence Alignment AGGCTATCACCTGACCTCCAGGCCGATGCCC TAGCTATCACGACCGCGGTCGATTTGCCCGAC -AGGCTATCACCTGACCTCCAGGCCGA--TGCCC--- TAG-CTATCAC--GACCGC--GGTCGATTTGCCCGAC Definition Given two strings x = x 1 x 2...x M, y = y 1 y 2 y N, an alignment is an assignment of gaps to positions 0,, N in x, and 0,, N in y, so as to line up each letter in one sequence with either a letter, or a gap in the other sequence

19 What is a good alignment? AGGCTAGTT, AGCGAAGTTT AGGCTAGTT- 6 matches, 3 mismatches, 1 gap AGCGAAGTTT AGGCTA-GTT- 7 matches, 1 mismatch, 3 gaps AG-CGAAGTTT AGGC-TA-GTT- AG-CG-AAGTTT 7 matches, 0 mismatches, 5 gaps

20 Scoring Function Sequence edits: Mutations Insertions Deletions Scoring Function: Match: +m Mismatch: -s Gap: -d AGGCCTC AGGACTC AGGGCCTC AGG. CTC Alternative definition: minimal edit distance Given two strings x, y, find minimum # of edits (insertions, deletions, mutations) to transform one string to the other Score F = (# matches) m - (# mismatches) s (#gaps) d

21 How do we compute the best alignment? AGTGCCCTGGAACCCTGACGGTGGGTCACAAAACTTCTGGA AGTGACCTGGGAAGACCCTGACCCTGGGTCACAAAACTC Too many possible alignments: (exercise) >> 2 N

22 Alignment is additive Observation: The score of aligning x 1 x M y 1 y N is additive Say that x 1 x i x i+1 x M aligns to y 1 y j y j+1 y N The two scores add up: F(x[1:M], y[1:n]) = F(x[1:i], y[1:j]) + F(x[i+1:M], y[j+1:n])

23 Dynamic Programming There are only a polynomial number of subproblems Align x 1 x i to y 1 y j Original problem is one of the subproblems Align x 1 x M to y 1 y N Each subproblem is easily solved from smaller subproblems We will show next Then, we can apply Dynamic Programming!!! Let F(i, j) = optimal score of aligning x 1 x i y 1 y j F is the DP Matrix or Table Memoization

24 Dynamic Programming (cont d) Notice three possible cases: 1. x i aligns to y j x 1 x i-1 x i y 1 y j-1 y j F(i, j) = F(i 1, j 1) + m, if x i = y j -s, if not 2. x i aligns to a gap x 1 x i-1 x i y 1 y j - F(i, j) = F(i 1, j) d 3. y j aligns to a gap x 1 x i - y 1 y j-1 y j F(i, j) = F(i, j 1) d

25 Dynamic Programming (cont d) How do we know which case is correct? Inductive assumption: F(i, j 1), F(i 1, j), F(i 1, j 1) are optimal Then, F(i, j) = max F(i 1, j 1) + s(x i, y j ) F(i 1, j) d F(i, j 1) d Where s(x i, y j ) = m, if x i = y j ; -s, if not

26 Example x = AGTA m = 1 Procedure to output y = ATA s = -1 Alignment d = -1 F(i,j) i = A G T A j = A T Follow the backpointers F(1, 1) = When max{f(0,0) diagonal, + s(a, A), OUTPUT x F(0, 1) i, y j d, F(1, 0) d} = When up, OUTPUT max{0 + 1, y j -1 1, When left, -1 1} = 1 OUTPUT x i 3 A A A G - T T A A

27 The Needleman-Wunsch Matrix x 1 x M y 1 y N Every nondecreasing path from (0,0) to (M, N) corresponds to an alignment of the two sequences An optimal alignment is composed of optimal subalignments

28 The Needleman-Wunsch Algorithm Initialization. F(0, 0) = 0 F(0, j) = - j d F(i, 0) = - i d Main Iteration. Filling-in partial alignments For each i = 1 M For each j = 1 N F(i 1,j 1) + s(x i, y j ) [case 1] F(i, j) = max F(i 1, j) d [case 2] F(i, j 1) d [case 3] DIAG, if [case 1] Ptr(i, j) = LEFT, if [case 2] UP, if [case 3] 3. Termination. F(M, N) is the optimal score, and from Ptr(M, N) can trace back optimal alignment

29 Performance Time: O(NM) Space: O(NM) Later we will cover more efficient methods

30 A variant of the basic algorithm: Maybe it is OK to have an unlimited # of gaps in the beginning and end: CTATCACCTGACCTCCAGGCCGATGCCCCTTCCGGC GCGAGTTCATCTATCAC--GACCGC--GGTCG Then, we don t want to penalize gaps in the ends

31 Different types of overlaps Example: 2 overlapping reads from a sequencing project Example: Search for a mouse gene within a human chromosome

32 The Overlap Detection variant x 1 x M Changes: y 1 y N 1. Initialization For all i, j, F(i, 0) = 0 F(0, j) = 0 2. Termination max i F(i, N) F OPT = max max j F(M, j)

33 The local alignment problem Given two strings x = x 1 x M, y = y 1 y N Find substrings x, y whose similarity (optimal global alignment value) is maximum x = aaaacccccggggtta y = ttcccgggaaccaacc

34 Why local alignment Genes are shuffled between genomes

35 Cross-species genome similarity 98% of genes are conserved between any two mammals >70% average similarity in protein sequence hum_a : 57331/ mus_a : 78560/ rat_a : / fug_a : 36008/68174 hum_a : 57381/ mus_a : 78610/ rat_a : / fug_a : 36058/68174 hum_a : 57431/ mus_a : 78659/ rat_a : / fug_a : 36084/68174 atoh enhancer in human, mouse, rat, fugu fish hum_a : 57481/ mus_a : 78708/ rat_a : / fug_a : 36097/68174

36 The Smith-Waterman algorithm Idea: Ignore badly aligning regions Modifications to Needleman-Wunsch: Initialization: F(0, j) = F(i, 0) = 0 Iteration: F(i, j) = max F(i 1, j) d 0 F(i, j 1) d F(i 1, j 1) + s(x i, y j )

37 The Smith-Waterman algorithm Termination: 1. If we want the best local alignment F OPT = max i,j F(i, j) Find F OPT and trace back 2. If we want all local alignments scoring > t?? For all i, j find F(i, j) > t, and trace back? Complicated by overlapping local alignments Waterman Eggert 87: find all non-overlapping local alignments with minimal recalculation of the DP matrix

38 Scoring the gaps more accurately Current model: γ(n) Gap of length n incurs penalty n d However, gaps usually occur in bunches Concave gap penalty function γ(n) (aka Convex -γ(n)): γ(n) γ(n): for all n, γ(n + 1) - γ(n) γ(n) - γ(n 1)

39 Convex gap dynamic programming Initialization: same Iteration: F(i 1, j 1) + s(x i, y j ) F(i, j) = max max k=0 i-1 F(k, j) γ(i k) max k=0 j-1 F(i, k) γ(j k) Termination: same Running Time: O(N 2 M) (assume N>M) Space: O(NM)

40 Compromise: affine gaps γ(n) = d + (n 1) e gap gap open extend To compute optimal alignment, γ(n) d e At position i, j, need to remember best score if gap is open best score if gap is not open F(i, j): score of alignment x 1 x i to y 1 y j if x i aligns to y j G(i, j): score if x i aligns to a gap after y j H(i, j): score if y j aligns to a gap after x i V(i, j) = best score of alignment x 1 x i to y 1 y j

41 Needleman-Wunsch with affine gaps Why do we need matrices F, G, H? Because, perhaps x i aligns to y j G(i, j) < V(i, j) x 1 x i-1 x i x i+1 y 1 y j-1 y j - (it is best to align x i to y j if we were aligning only x 1 x i to y 1 y j and not the rest of x, y), but on the contrary Add -d G(i+1, j) = F(i, j) d G(i, j) x e i aligns > V(i, j) to d a gap after y j x 1 x i-1 x i x i+1 y 1 y j - - (i.e., had we fixed our decision that x i aligns to y j, we could regret it at the next step when aligning x 1 x i+1 to y 1 y j ) Add -e G(i+1, j) = G(i, j) e

42 Needleman-Wunsch with affine gaps Initialization: V(i, 0) = d + (i 1) e V(0, j) = d + (j 1) e Iteration: V(i, j) = max{ F(i, j), G(i, j), H(i, j) } F(i, j) = V(i 1, j 1) + s(x i, y j ) G(i, j) = max V(i 1, j) d G(i 1, j) e Termination: V(i, j 1) d H(i, j) = max H(i, j 1) e V(i, j) has the best alignment Time? Space?

43 To generalize a bit think of how you would compute optimal alignment with this gap function γ(n).in time O(MN)

44 Bounded Dynamic Programming Assume we know that x and y are very similar Assumption: # gaps(x, y) < k(n) x i Then, implies i j < k(n) y j We can align x and y more efficiently: Time, Space: O(N k(n)) << O(N 2 )

45 Bounded Dynamic Programming y 1 y N x 1 x M k(n) Initialization: F(i,0), F(0,j) undefined for i, j > k Iteration: For i = 1 M For j = max(1, i k) min(n, i+k) F(i 1, j 1)+ s(x i, y j ) F(i, j) = max F(i, j 1) d, if j > i k(n) F(i 1, j) d, if j < i + k(n) Termination: same Easy to extend to the affine gap case

46 Outline Linear-Space Alignment BLAST local alignment search Ultra-fast alignment for (human) genome resequencing

47 Linear-Space Alignment

48 Subsequences and Substrings Definition A string x is a substring of a string x, if x = ux v for some prefix string u and suffix string v (similarly, x = x i x j, for some 1 i j x ) A string x is a subsequence of a string x if x can be obtained from x by deleting 0 or more letters (x = x i1 x ik, for some 1 i 1 i k x ) Note: a substring is always a subsequence Example: x = abracadabra y = cadabr; substring z = brcdbr; subseqence, not substring

49 Hirschberg s algortihm Given a set of strings x, y,, a common subsequence is a string u that is a subsequence of all strings x, y, Longest common subsequence Given strings x = x 1 x 2 x M, y = y 1 y 2 y N, Find longest common subsequence u = u 1 u k Algorithm: F(i 1, j) F(i, j) = max F(i, j 1) F(i 1, j 1) + [1, if x i = y j ; 0 otherwise] Ptr(i, j) = (same as in N-W) Termination: trace back from Ptr(M, N), and prepend a letter to u whenever Ptr(i, j) = DIAG and F(i 1, j 1) < F(i, j) Hirschberg s original algorithm solves this in linear space

50 Introduction: Compute optimal score It is easy to compute F(M, N) in linear space Allocate ( column[1] ) Allocate ( column[2] ) F(i,j) For If i = 1.M i > 1, then: Free( column[ i 2 ] ) Allocate( column[ i ] ) For j = 1 N F(i, j) =

51 Linear-space alignment To compute both the optimal score and the optimal alignment: Divide & Conquer approach: Notation: x r, y r : reverse of x, y E.g. x = accgg; x r = ggcca F r (i, j): optimal score of aligning x r 1 xr i & yr 1 yr j same as aligning x M-i+1 x M & y N-j+1 y N

52 Linear-space alignment Lemma: (assume M is even) F(M, N) = max k=0 N ( F(M/2, k) + F r (M/2, N k) ) x y M/2 F(M/2, k) F r (M/2, N k) k * Example: ACC-GGTGCCCAGGACTG--CAT ACCAGGTG----GGACTGGGCAG k * = 8

53 Linear-space alignment Now, using 2 columns of space, we can compute for k = 1 M, F(M/2, k), F r (M/2, N k) PLUS the backpointers x 1 x M/2 x M x 1 x M/2+1 x M y 1 y 1 y N y N

54 Linear-space alignment Now, we can find k * maximizing F(M/2, k) + F r (M/2, N-k) Also, we can trace the path exiting column M/2 from k * 0 1 M/2 M/2+1 M M+1 k * k * +1

55 Linear-space alignment Iterate this procedure to the left and right! k * N-k * M/2 M/2

56 Linear-space alignment Hirschberg s Linear-space algorithm: MEMALIGN(l, l, r, r ): (aligns x l x l with y r y r ) 1. Let h = (l -l)/2 2. Find (in Time O((l l) (r r)), Space O(r r)) the optimal path, L h, entering column h 1, exiting column h Let k 1 = pos n at column h 2 where L h enters k 2 = pos n at column h + 1 where L h exits 3. MEMALIGN(l, h 2, r, k 1 ) 4. Output L h 5. MEMALIGN(h + 1, l, k 2, r ) Top level call: MEMALIGN(1, M, 1, N)

57 Linear-space alignment Time, Space analysis of Hirschberg s algorithm: To compute optimal path at middle column, For box of size M N, Space: 2N Time: cmn, for some constant c Then, left, right calls cost c( M/2 k * + M/2 (N k * ) ) = cmn/2 All recursive calls cost Total Time: cmn + cmn/2 + cmn/4 +.. = 2cMN = O(MN) Total Space: O(N) for computation, O(N + M) to store the optimal alignment